Using linear law with real data
In experiments the variables often follow a non-linear law such as y = axn or y = abx. By transforming the data to the form Y = mX + c and drawing a line of best fit, we estimate the constants even when the readings contain small errors.
Steps
- Choose X and Y so that the law becomes Y = mX + c.
- Tabulate X and Y from the data and plot the points.
- Draw the best straight line and read m and c.
- Recover the original constants (for example a = 10c, n = m).
Because experimental readings are never perfect, the plotted points rarely lie on an exact line. Drawing a single straight line that passes as close as possible to all of the points averages out these random errors and gives more reliable constants than using any two readings on their own. After finding a and n (or b), substitute a chosen value of x back into the original law to predict y, or rearrange the law to predict x from a given y.
Worked example
An experiment follows y = abx. A plot of log y against x is a straight line through (0, 0.30) and (2, 0.90). Gradient = (0.90 − 0.30)/(2 − 0) = 0.30 = log b, so b = 100.30 ≈ 2.00. Intercept = 0.30 = log a, so a = 100.30 ≈ 2.00. The law is y ≈ 2.00(2.00)x.
Key formula
From the best-fit line Y = mX + c:
gradient m = (Y₂ − Y₁)/(X₂ − X₁)
for y = abx: b = 10m, a = 10c
Remember
- Use a line of best fit — do not join dots.
- Read the intercept where X = 0, if it lies on the axis.
- Predictions use the recovered constants in the original law.